Point force contacts

Once a specific event in the load-displacement curve has been associated with a particular phase transition, the pressure at which it occurs can be estimated by considering the elastoplastic behavior of the material under the indenter. For the point-force contact, the Sneddon s solution [39] to the problem of the penetration of an axisymmetric punch into an elastic half space predicts the following relation between the applied load P and the indenter displacement A ... 

The pores in question can represent only a small fraction of the pore system since the amount of enhanced adsorption is invariably small. Plausible models are solids composed of packed spheres, or of plate-like particles. In the former model, pendulate rings of liquid remain around points of contact of the spheres after evaporation of the majority of the condensate if the spheres are small enough this liquid will lie wholly within the range of the surface forces of the solid. In wedge-shaped pores, which are associated with plate-like particles, the residual liquid held in the apex of the wedge will also be under the influence of surface forces. 

In the pendular state, shown in Figure la, particles ate held together by discrete lens-shaped rings at the points of contact or near-contact. For two uniformly sized spherical particles, the adhesive force in the pendular state for a wetting Hquid (contact angle zero degree) can be calculated (19,23) and substituted for H. in equation 1 to yield the foUowing, where y is the Hquid surface tension in N/m. 

The frictional behaviour of rubber is quite different from that of metals. In Chapter 25 we showed that when metallic surfaces were pressed together, the bulk of the deformation at the points of contact was plastic and that the friction between the surfaces arose from the forces needed to shear the junctions at the areas of contact. 

Figure 26.49 (upper) shows the elastic stored energy of two hne forces acting at an angle a to the surface of the semi-inhnite body simultaneously and Figure 26.49 (lower) the horizontal stress component. As expected both energy and stress show strong peaks at the points of contact. 

FIGURE 6.8 Three force curves taken at locations of a gelatin film thickness of 150 nm (curve A), 410 nm (curve B), and 1.15 pm (curve C). At high forces, the force curves are steeper for small thicknesses because the cantilever deflection is influenced by the underlying stiff substrate at these small film thicknesses. For comparing the slopes more easily, the curves are shifted such that their points of contact coincide. Reprinted with permission from Domke and Radmacher (1998). 

It is instructive to simplify the above picture somewhat and consider the coalescence or sticking of two particles schematically shown in Fig. 13. One can assume that due to shear forces in the mixer, a fluidized bed in the present case, the two particles posses a relative velocity U0 which ensures collision at some point on their trajectory and possible sticking under appropriate conditions. It is essential that some binder be present at the point of contact, as depicted in the figure. From this simplified picture, allmechanisms... 

The net area of this intimate contact is called the real area of contact Areai. It is assumed that plastic flow occurs at most microscopic points of contact, so that the normal, local pressures correspond to the hardness aj, of the softer of the two materials that are in contact. The (maximum) shear pressure is given by the yield strength cry of the same material. The net load L and the net shear force Fs follow by integrating aj, and cry over the real area of contact Areai. That is, L = cs, Arca and Fs = ayAreai. Hence, the plastic deformation scenario results in the following (static) friction coefficient ... 

Formation of agglomerates by powder compaction may involve rearrangement of particles to increase their packing efficiency resulting in the enhancement of interparticle adhesion forces [89]. Furthermore, particle deformation at the point of contact between particles can greatly increase both the contact surface area and interparticle attraction [84]. 

A Models to describe microparticles with a core/shell structure. Diametrical compression has been used to measure the mechanical response of many biological materials. A particular application has been cells, which may be considered to have a core/shell structure. However, until recently testing did not fully integrate experimental results and appropriate numerical models. Initial attempts to extract elastic modulus data from compression testing were based on measuring the contact area between the surface and the cell, the applied force and the principal radii of curvature at the point of contact (Cole, 1932 Hiramoto, 1963). From this it was possible to obtain elastic modulus and surface tension data. The major difficulty with this method was obtaining accurate measurements of the contact area. 

During a sliding movement the points of contact, where the two surfaces are welded together, are continuously being broken, whereafter they reform this requires a force %-A, in which % is the shear strength of the material. This is the friction force, F = %-A = p-N, where p is the coefficient of friction. This leads to a very simple expression for p ... 

Cohesion. This test is used for very fine powders (below 70 pm). Material is passed through three vibrating sieves in series. The material left on each sieve is weighed and a cohesion index is determined from the relative amounts retained. Carr19 defined cohesion as the apparent surface force acting on the surface of powders, which are composed of millions of atoms. The number of points of contact within the powder mass determines the effect of this force. Thus, cohesiveness increases with decreasing particle size, since the number of contact points increases as the particle size decreases. 

Attraction of Two Spheres—An expression for the attraction between two or more particles, based on collision theory rather than attraction due to the motion of spheres, may be developed by the methods of dimensional analysis. Let the force of attraction between two particles of diameters d and d2 be F and assume that when the particles are close enough so that the gaseous film enveloping each particle coalesces over a region about the point of contact, then the whole attraction is due to a free surface energy a. If the average surface of contact of the particles is denoted by Sc then... 

Fig. 2. A. The force between silica surfaces at cg= 1M and pH=9, determined experimentally in Ref. [22] (circles) the continuous line represents an unsuccessful fit with Eq. (43a). The origin of the x axis corresponds to the smallest separation distance between the two surfaces, attainable by AFM. The coordinate of the true point of contact between the surfaces cannot be obtained directly from experiment. The inset shows the region of the secondary minimum, where the magnitude of the interaction is comparable with instrumental resolution of 0.01 mN/m, B. The fit of the experimental data (cE=- 1M, pH=9) with an exponential repulsion and a van der Waals attraction (Eq. (43b)). As in Fig. 2A, the origin of the t axis corresponds to the smallest separation attained in the experiment. The true point of contact between surfaces, obtained from fit, is located at a distance which is by 2 =15 A larger than the smallest separation recorded by AFM.

Other forces existing between particles often operate against hydrodynamic drag that produces fluidization. Thus, when particles touch one another, there exists a London-van der Waals force of a molecular nature at the point of contact. This force looms in proportion when gravity and drag forces diminish as a particle becomes smaller. For a small particle, the surface force of adhesion may often be thousands of times greater than its weight. 

When a force is applied to a material, deformation occurs. When this deformation completely disappears after cessation of the external force, further deformation occurs. Deformations that do not completely recover after release of the stress are known as plastic deformations. The force required to initiate a plastic deformation is known as the yield stress. When the particles are so closely packed that no further filling of the voids can occur, a further increase of the compressional force causes deformation at the points of contact. Both plastic and elastic deformation may occur,... 

The wettability theory of adhesion is inextricably related to the study of contact angles of liquids on solid surf aces. A force balance at the point of contact between the liquid and the solid can be written (3)... 

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